Consider a function of two variables $f(x,y)$. I am interested in solving the recurrence relation
$$a \left(f(x,y-1)-f(x,y+1)\right)=b \left(f(x-1,y)-f(x+1,y)\right)$$
for a most general $f(x,y)$, where $a$ and $b$ are constants.
Unfortunately, I am not completely sure how to approach this problem. Do some techniques exist that would allow to tackle this task? What would be a fruitful way to proceed?